From the works of Borel and Quillen there is a connection between the $K$-theory of the ring of integers $\mathfrak{o}_K$ in a number field $K$ and the arithmetic of the number field. In fact, it is well known that

$K_0(\mathfrak{o}_K)=\mathbb{Z}\oplus \mathrm{Cl}(\mathfrak{o}_K)$, where $\mathrm{Cl}(\mathfrak{o}_K)$ denotes the class group of $K$;

$K_1(\mathfrak{o}_K)=\mathfrak{o}_K^\times$ (group of units in $\mathfrak{o}_K$);

For $n\geq 2$ all the $K$-groups are finitely generated:

$K_{2n}(\mathfrak{o}_K)$ is a finite group for $n\geq 1$;

$K_{2n-1}(\mathfrak{o}_K)= \mathbb{Z}^{d_n}\oplus (\text{finite group})$ where $d_n=r_1+r_2$ or $r_2$ (the number of real/complex embeddings), depending on if $n\equiv 1 \mod 4$ or $n\equiv 3\mod 4$.

Also, and here comes the pretty remarkable meat of the story: for $n=2m-1$, the order of vanishing of the Dedekind $\zeta$-function, $\zeta_K(s)$, at $s=1-m$ is equal to the rank of $K_{2m-1}$.

(All this can be found in Weibel's survey in the $K$-theory handbook, or Kolster's survey from ICTP; google and ye shall find.)

Now, my questions are:

1) Since the definitions of the $K$-groups are pretty topological in nature, in addition quite complicated (and I am very far from being a topologist), is there any 'reason' why this connection between $K$-theory and the arithmetic of number fields exists at all. It seems like a complete miracle. In other words, why should this connection exist?

2) What is the arithmetic significance of the torsion parts of the above $K$-groups? There is some information on this in Weibel's survey mentioned above, but I don't understand what he is talking about.

Admittedly, question 1) is vague, but I hope it's clear what I'm after.

There is much more to your bullet point 3 than you indicate. Kolster's survey that you reference has lots of good information about this. For instance, see the Lichtenbaum conjecture on p.199, which is, to your third bullet point, what the strong BSD conjecture about the lead coefficient at s=1 of the L-function of an elliptic curve is to the weak version that only describes the order of vanishing. As for other arithmetic significance, you can easily define $K_2$ arithmetically. It is sometimes called the "tame kernel" (which you might google), defined as the kernel of a certain symbol.
–
BarryJul 13 '11 at 12:50

1

As far as you question 1 goes, the better way to think about why the connection between K-groups and L-functions exists is by viewing the K-groups as "almost" being etale cohomology groups (i.e., identical up to the 2-torsion part).
–
BarryJul 13 '11 at 13:14

@Barry: Yep, I knew this, but I think the above is sufficient motivation for my questions. As for your second comment, then the question becomes, why are the $K$-groups almost étale cohomology. In other words, I don't think your suggestion is enough (for me at least) but maybe a distance in the right direction.
–
Daniel LarssonJul 13 '11 at 13:49

8

Shifting attention to a different aspect of your question 1: Why should there be an important invariant in algebra that requires topology in its definition? Well, in algebra we are accustomed to thinking about generators, relations, and relations between relations, etc. That's homological algebra. In that spirit, $K_0$ is about whether a module has a basis, $K_1$ is about whether two bases are the same (up to basic row-reduction moves), $K_2$ is whether two ways of getting from one basis to another by such moves are the same up to .... Homotopy theory is a good way of organizing these ideas.
–
Tom GoodwillieJul 13 '11 at 17:54

@Tom: that's certainly a way to think about it, but in my mind it leads to the question: thinking of homotopy theory (once again, I'm not a topoplogist), $\pi_n$ are the natural objects of study (and so we're led to Quillen's definition of $K$-groups as I see it), and then this would lead more into Galois representations as the Galois group is sort of the fundamental group of the number field. On the other hand, come to think of it, the Artin $L$-function involves a representaion of the Galois group. Hmmm... you could be onto something...
–
Daniel LarssonJul 14 '11 at 6:47